Rotational Motion & Equilibrium

AP Physics Rapid Learning Series - 10

© Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 1

Rapid Learning CenterChemistry :: Biology :: Physics :: Math

Rapid Learning Center Presents …Rapid Learning Center Presents …

Teach Yourself AP Physics in 24 Hours

1/53*AP is a registered trademark of the College Board, which does not endorse, nor is

affiliated in any way with the Rapid Learning courses.

Rotational MotionRotational Motion and Equilibrium

Physics Rapid Learning Series

2/53

Rapid Learning Centerwww.RapidLearningCenter.com/

© Rapid Learning Inc. All rights reserved.

Wayne Huang, Ph.D.Keith Duda, M.Ed.

Peddi Prasad, Ph.D.Gary Zhou, Ph.D.

Michelle Wedemeyer, Ph.D.Sarah Hedges, Ph.D.



Learning Objectives

Describe the kinematics of rotational motion.

By completing this tutorial, you will:

Understand the concept of torque.Understand the concept of moment of inertiaApply the concepts of torque and moment of inertia to rotational motion

3/53

inertia to rotational motion and equilibrium.Extend your knowledge of momentum to the rotational variety.

Concept MapPhysics

Studies

Previous content

New content

Motion

Caused byRotational

Motion

Linear Motion

Possesses Angular speed

4/53

Torques

Caused byForces

Angular momentum

May be inEquilibrium



Kinematics ofKinematics of Rotational Motion

Our previous knowledge of kinematics and motion in a line can be extended to

5/53

and motion in a line can be extended to describe rotational motion.

Angular Analogs

When linear distance, d, was measured, we used meters or similar metric unit.similar metric unit.

Now we must use radians to measure angular or rotational distance, θ.

Thus a radian will be analogous

6/53

Thus a radian will be analogous to a meter, and angle will be analogous to distance for this tutorial.



The Radian

Angle, radians

Linear arc distance

traveled, m

rsθ =

θ

7/53

Radius of circular path, m

Note that meters and meters cancels out. Thus a radian has no other dimensional units.

Radians and Circles

A useful fact to remember is that one complete circle equals 2π radians.

If a wheel turns 200 rpm, revolutions per minute, how many radians per second is that?

irev200

1rad 2πx

601minx rad/sec20.9=

8/53

min 1rev 60s



Angular Velocity

Just as there is a rotational equivalent to linear distance, there is one for linear speed.

∆t∆θω =

Angle, radAngular speed, rad/s

9/53

∆t Time,s

This is comparable to our linear speed formula:

V = d/t

Angular Acceleration

Just as there is a rotational equivalent to linear velocity, there is one for linear acceleration.

Ch i

∆t∆ωα =

Change in angular speed,

radians/s

Angular acceleration,

rad/s2

10/53

∆t Time,s

This is comparable to our linear acceleration formula: a = ∆v/∆t



Additional Angular FormulasWe already have two definitions or formulas for angular motion that are similar to previous linear ones.

Linear Motion Angular Motion

∆t∆ωα =

∆t∆θω =

tdv =

∆t∆va =

/2ttd 2 /2ttθ 2

11/53

There are corresponding formulas for our other linear relationships too.

/2attvd 2i += /2αttωθ 2

i +=

2advv 2i

2f += θ2αωω 2

i2f +=

Angular Motion Example - Question

The blade in an electric circular saw starts at rest. When initiated, the blade reaches an angular velocity of 90 rev/s in 250 revolutions. What is the angular acceleration of the blade?

12/53



Angular Motion Example - Solution

Given quantities:

ωi = 0 rad/s ωf = 90 rev/s θ = 250 rev

Unknown quantity:

angular acceleration, α

Likely formula to use:

13/53

ωf2 =ωi

2 + 2αθ

Rearranging for unknown quantity:

2θωωα

2i

2f −=

Angular Motion Example - Calculation

First, units must be converted:

frev 2π radω 90 x 180π rad/s= =f s 1rev

2π radθ 250rev x 500π rad1rev

= =

Next, substitute into our rearranged formula:22

14/53

2θωωα

2i

2f −=

2 22(180π rad/s) (0π rad/s)α 102 rad/s

2(500π rad)−

= =



Torque

Just as force relates to linear movement, torque has an analogous relationship to

15/53

torque has an analogous relationship to rotation.

Just as force makes objects move in a line, torque makes object rotate.

Ƭ F l

Torque Formula

Ƭ = F l

Torque, mN

Force applied, N

Lever arm distance,

m

16/53

m

The notation can be a bit confusing:

Don’t confuse Ƭ for torque with T for time period. Don’t confuse l for lever arm with a number 1.



Torque UnitsAs the formula implies, torque has typical metric units of N·m.

You may notice this is very similar to the unit of y yJoules for work/energy which may be written as N·m.

17/53

To avoid confusion, the unit of torque is often written reversed: m·N

Direction of Force

When calculating torque, only the force that is perpendicular, ┴ , to the lever arm is considered.

Pushing along, or parallel, to the lever arm doesn’t produce any torque!

Lever armpivot

18/53

Some torque More torque since all force is ┴to lever arm.

No torque since no force is ┴ to lever arm.



Increasing TorqueWhen a lot of torque is required to tighten or loosen something, you can either increase your force applied, or increase the lever arm.

This is why many tools have large lever arms or handles to held accomplish the job:

19/53

Torque Wrench

Often, a bolt or other mechanical fastener must be tightened a certain amount. Too loose and the object falls off, too tight and the bolt may snap! To accomplish this, a torque wrench measures exactly that! There is no guess work involved then.

For the particular

20/53

For the particular lever arm, the dial reads the amount of torque applied.



Torque Example - Question

If you apply a 100 N force to the end of the wrench as shown, how much torque are you applying?

.50m45o

21/53

100N

Torque Example - Solution

Remember that we’re only using the component of the force that is perpendicular to the lever arm:

.50m45o

Ƭ = l F

Ƭ = l F sinθ

hypF

hypoppsinθ ⊥==

22/53

100NƬ = 36 m·N

Ƭ = .50m 100N sin45o

This vector represents the component of the applied force that is perpendicular to the lever arm



Torque Direction

Torque can have two directions:

Clockwise Counterclockwise

23/53

Clockwise (CW)

Counterclockwise (CCW)

When there is an unbalanced torque, rotation is caused. This is analogous to an unbalanced force causing acceleration.

Application of TorqueYou’ve probably experienced torque in the most common of circumstances. For example, every time you open a door, you apply a torque to the handle or knob The knob exists because ithandle or knob. The knob exists because it supplies a larger lever arm for you to apply more torque to the door mechanism.

Close-up of knob

24/53

Force applied, then rotation occurs.



Moment of Inertia

Also called rotational inertia, this is the spinning counterpart to linear inertia that

25/53

spinning counterpart to linear inertia that you are already familiar with.

Linear Inertia Review

Linear inertia says that an object moving in a straight line wants to continue moving in a straight line, until acted upon by a force.

Also, an object at rest wants to stay at rest.

Linear inertia is dependent upon mass.

26/53

Large amount of inertia!

Small amount of inertia!



Moment of Inertia

Moment of inertia says that an object rotating about an axis wants to continue rotating until acted on by a force, or torque.

27/53

Moment of inertia, or rotational inertia depends on the distribution of the mass in the rotating object.

Although mass is important, so is the location of that mass.

Distribution of Mass

When the mass is nearer the axis of rotation, it is easier to move, low rotational inertia.

m m

When the mass is farther from the axis, it is harder to move, high rotational inertia.

mass

mass

28/53

move, high rotational inertia.

mass

mass



Conceptual Example

A person balancing on a tightrope or other tenuous object may put their arms outward to balance. They may even carry objects to help them stay balanced.

29/53

This extra mass far away from them increases their moment of inertia. This makes it more difficult for them to rotate, or tip over and fall.

Moment of Inertia FormulaIf you consider any object to be made up of a collection of particles

I = Σmr2

Moment of inertia

The sum of all

Mass of particle

Radius

30/53

of inertia, kg•m2

of all particles

particle, kg

from axis of rotation, m

This odd formula describes how difficult it is to make an object rotate.



Other Shortcut FormulasThere are many formulas for finding the moment of inertia for various shapes. These can be found in any physics text. For example:

31/53

Solid disk

I = 1/2 mr2

Solid sphere

I = 2/5 mr2

Moment of Inertia - Example

Calculate the moment of inertia of a saw blade. The blade has a diameter of 25 cm, and a mass of 0.70 kg.

25 cm

32/53

Consider the blade a regular flat disk, neglecting the teeth and middle hole.



Moment of Inertia - SolutionChange the given diameter into a radius and convert into meters:

125m1mx12 5cm225cm ==÷

Since we are considering the blade a solid disk, use the moment of inertia shortcut formula:

2mrI

2

=

.125m100cm

x12.5cm2 25cm ==÷

33/53

2

22

.005kgm2

5m)(.7kg)(.12I ==

Mass and Inertia - Question

A disk, and a hoop have equal total masses. They are simultaneously released at the top of a hill. Which will hit the bottom first?

???

Hint: Consider where the mass is located on each.

34/53



Mass and Inertia - Answer

Although they both have the same mass, thus the same pull from gravity, the hoop has a higher moment of inertia since its mass is located farther from its axis of rotationfrom its axis of rotation.

The disk has some of its mass nearer the axis of rotation, thus a lower moment of inertia.

35/53

A lower moment of inertia means it is easier for a torque to get the object rotating.

Thus, the disk will win the race every time!

Rotational Motion and Equilibrium

Just as forces balance out to produce equilibrium torques may cancel out to

36/53

equilibrium, torques may cancel out to produce equilibrium too.



Balancing TorquesWhen the CW and CCW torques equal each other, then the object is in equilibrium. There is no rotation. Imagine this see-saw example shown belowbelow.

eclockwisckwisecounterclo ττ =ll FF =

)3N(4m2N(6m) =12Nm12Nm =

37/53

No rotation in this case: rotational equilibrium

Rotational Equilibrium

If the torques do balance out, and there are no other net forces acting, the system would be in equilibrium.

Obviously the forcesdo not need to balance out, but the torques do This is

38/53

torques do. This is true because torque is the product of a force and a distance.



Newton’s Second Law Again

When torques aren’t in equilibrium, rotation occurs. This is described by the rotational version of Newton’s second law :

IαΣ =τThe sum of

the torques

action on

Angular acceleration,

rad/s2

Newton s second law :

39/53

an object, N•m

Moment of inertia,

kg•m2

Torque CalculationUsing Newton’s second law in rotational form, calculate the torque needed to accelerate the saw blade. Use the data from previous example problemsproblems.

2.005kgmI =2rad/s 51α =

IαΣ =τ

Newton’s second law

Rad unit drops out

40/53

))(51rad/s(.005kgmΣ 22=τ

Nm 255/s.255kgmΣ 22 .==τ1N = kg•m/s2



Angular Momentum

Your previous knowledge of linear momentum can be transferred to the

41/53

momentum can be transferred to the angular arena.

Angular Momentum Description

Just as an object moving in a straight line has linear momentum, an object moving in a circular path has angular momentum.

Angular momentum is a measure of the “strength” of an object’s rotation about a particular axis.

42/53



Angular Momentum FormulaAngular momentum is the product of two rotational quantities we learned earlier in this tutorial:

L = I ωAngular momentum,

kg•m2/sMoment

of inertia, kg•m2

Angular speed, rad/s

43/53

kg•m

Notice the very unusual unit for angular momentum! Calculations of angular momentum can become very cumbersome!

SimilaritiesThe formula for linear momentum is analogous to the formula for angular momentum:

mvP = mvP =Moment

mass

Ve

l oc

44/53

IωL =um

i t y



Conservation of Angular Momentum

Just as linear momentum is conserved, so is angular momentum.

An object will maintain its angular momentum unless acted on by an unbalanced torque.

45/53

This is analogous to how an unbalanced force changes an object’s linear momentum.

Stability

Often, objects are rotated so that they are more stable.

Because they have angular momentum, it is more difficult to change their motion.

This is why a football is thrown with a spiral

46/53

motion.



The Gyroscope

A gyroscope is simply a rotating object that has significant angular momentum.

Like a rotating football, that momentum is difficult to change, so the gyroscope is very stable.

47/53

Helicopters and Momentum

Angular momentum is conserved by helicopters too. The main rotating blade has angular momentum. To keep momentum conserved, the

fbody of the helicopter should rotate the opposite way!

48/53

The tail rotor counteracts this tendency and keeps the helicopter steady.



Momentum Concept ExampleImagine a rotating horizontal disk similar to a record player. It spins at some particular angular speed.

m m

49/53

As it is spinning, additional masses are added to the disk. How will the situation change?

Changing QuantitiesIn the previous example, the angular speed obviously decreased.

This occurred because of the additional mass

angular moment angular

This occurred because of the additional mass which raised the moment of inertia.

However, combining the two, shows that the angular momentum is conserved and stays constant.

50/53 IωL =

speedof inertiamomentum

constant



The torque on an item equals the moment of

The torque on an item equals the moment of

All linear kinematics

equations can

All linear kinematics

equations can

All previous concepts

involving linear momentum

All previous concepts

involving linear momentum

Learning Summary

Moment of Moment of

the moment of inertia times the angular

acceleration.

the moment of inertia times the angular

acceleration.

qbe translated into rotational

varieties

qbe translated into rotational

varieties

momentum hold true for the angular momentum.

momentum hold true for the angular momentum.

51/53

inertia depends upon the mass

of a item and its distribution of

that mass.

inertia depends upon the mass

of a item and its distribution of

that mass.

Torque is the product of force times lever arm.

Torque is the product of force times lever arm.

Congratulations

You have successfully completed the tutorial

Rotational Motion and EquilibriumEquilibrium

Rapid Learning Center



Rapid Learning Center

Wh t’ N t

Chemistry :: Biology :: Physics :: Math

What’s Next …

Step 1: Concepts – Core Tutorial (Just Completed)

Step 2: Practice – Interactive Problem Drill

Step 3: Recap Super Review Cheat Sheet

53/53

Step 3: Recap – Super Review Cheat Sheet

Go for it!

http://www.RapidLearningCenter.com

Education

Rotational Motion & Equilibrium